Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries

(1)

Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries

^∗

Dániel Marx

^†

School of Computer Science Tel Aviv University

Tel Aviv, Israel

dmarx@cs.bme.hu

ABSTRACT

An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints [19, 24]. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomial-time solvable or fixed-parameter tractable, parameterized by the number of variables. In the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameter- ization by the number of variables is a meaningful question.

The most general known property ofHthat makes CSP(H) polynomial-time solvable is bounded fractional hypertree width.

Here we introduce a new hypergraph measure calledsubmod- ular width, and show that bounded submodular width of H(which is a strictly more general property than bounded fractional hypertree width) implies that CSP(H) is fixed- parameter tractable. In a matching hardness result, we show that ifHhas unbounded submodular width, then CSP(H) is not fixed-parameter tractable (and hence not polynomial- time solvable), unless the Exponential Time Hypothesis (ETH) fails. The algorithmic result uses tree decompositions in a novel way: instead of using a single decomposition depend- ing on the hypergraph, the instance is split into a set of

†Research supported by ERC Advanced Grant DMMCA.

∗The full version of the paper is available at http://arxiv.org/abs/0911.0801

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

STOC’10, June 5–8, 2010, Cambridge, Massachusetts, USA.

instances (all on the same set of variables as the original instance), and then the new instances are solved by choosing a different tree decomposition for each of them. The reason why this strategy works is that the splitting can be done in such a way that the new instances are “uniform” with respect to the number extensions of partial solutions, and therefore the number of partial solutions can be described by a submodular function. For the hardness result, we prove via a series of combinatorial results that if a hypergraphH has large submodular width, then a 3SAT instance can be efficiently simulated by a CSP instance whose hypergraph is H. To prove these combinatorial results, we need to develop a theory of (multicommodity) flows on hypergraphs and vertex separators in the case when the function b(S) defining the cost of separatorS is submodular.

Categories and Subject Descriptors

F.2 [Theory of Computing]: Analysis of Algorithms and Problem Complexity

General Terms

Algorithms, Design, Performance, Theory

Keywords

constraint satisfaction, conjunctive queries, submodular width, fixed-parameter tractability

1. INTRODUCTION

There is a long line of research devoted to identifying hypergraph properties that make the evaluation of conjunctive queries tractable (see e.g. [15, 31, 18, 19]). In the early 80s, it has been noted that acyclicity is one such property [5, 11, 32, 4]. Later, more general such properties were iden- tified in the literature: for example, bounded query width [7], bounded hypertree width [15], and bounded fractional hypertree width [25, 20]. Our main contribution is giving a complete theoretical answer to this question: in a very precise technical sense, we characterize those hypergraph properties that imply tractability for the evaluation of a query. Efficient evaluation of queries is originally a question of database theory; however, the problem can be treated as a constraint satisfaction problem [23, 17, 31].

Constraint satisfaction. Constraint Satisfaction Prob- lems (CSP) form a general framework that includes many standard algorithmic problems such as satisfiability, graph

(2)

coloring, database queries, etc. [18, 12]. A CSP instance consists of a set V of variables, a domainD, and a set C of constraints, where each constraint is a relation on a subset of the variables. The task is to assign a value fromD to each variable in such a way that every constraint is satisfied. It is easy to see that Boolean Conjunctive Query can be formulated as the problem of deciding if a CSP instance has a solution: the variables of the CSP instance correspond to the variables appearing in the query and the constraints correspond to the database relations. A distinc- tive feature of CSP instances obtained this way is that the number of variables is small (as queries are typically small), while the domain of the variables are large (as the database relations usually contain a large number of entries). This has to be contrasted with typical CSP problems from AI, such as 3-colorability and satisfiability, where the domain is small, but the number of variables is large. As our motivation is database-theoretic, in the rest of the paper the reader should keep in mind that we are envisioning scenarios where the number of variables is small and the domain is large.

A natural question is to investigate the effect ofstructural restrictionson the complexity of CSP; that is, what kind of restrictions on the structure induced by the constraints on the variables make the problem tractable. Thehypergraphof a CSP instance is defined to be a hypergraph on the variables of the instance such that for each constraintc∈C there is a hyperedgeeccontaining exactly the variables appearingc.

If the hypergraph of the CSP instance has simple structure, then the instance is easy to solve. It is well-known that a CSP instanceI with hypergraph H can be solved in time kIk^O(tw(H)) [14], where tw(H) denotes the treewidth of H andkIkis the size of the representation ofI.

Our goal is to characterize the “easy” and “hard” hypergraphs from the viewpoint of constraint satisfaction. How- ever, formally speaking, CSP is polynomial-time solvable for every fixed hypergraphH: sinceHhas a constant number k of vertices, every CSP instance with hypergraph H can be solved by trying allkIk^k possible combinations on thek variables. It makes more sense to characterize thoseclasses of hypergraphs where CSP is easy. For a classHof hypergraphs, let CSP(H) be the restriction of CSP to instances whose hypergraphs are in H. For the characterization of the complexity of CSP(H), we can investigate two notions of tractability. CSP(H) ispolynomial-time solvableif there is an algorithm solving every instance of CSP(H) in time (kIk)^O(1), wherekIkis the length of the representation ofI in the input. For example, ifHis class of hypergraphs with bounded treewidth, then CSP(H) is polynomial-time solvable. The following notion interprets tractability in a less restrictive way: CSP(H) isfixed-parameter tractable (FPT) if there is an algorithm solving every instanceI of CSP(H) in timef(H)(kIk)^O(1), wheref is an arbitrary function.

The case of bounded arities. If the constraints have bounded arity (i.e., the edge size inHis bounded by a con- stantr), then CSP(H) is well understood: bounded treewidth is the only polynomial-time solvable case.

Theorem 1.1 ([19]). IfHis a recursively enumerable class of hypergraphs with bounded edge size, then (assuming FPT6= W[1]) the following are equivalent:

1. CSP(H) is polynomial-time solvable.

2. CSP(H) is fixed-parameter tractable.

3. Hhas bounded treewidth.

Theorem 1.1 proves the surprising result that whenever CSP(H) is fixed-parameter tractable, it is polynomial-time solvable as well. The following sharpening of Theorem 1.1 shows that there is no algorithm whose running time is signifi- cantly better than the kIk^O(tw(H)) bound of the treewidth based algorithm. The result is proved under the Exponential Time Hypothesis (ETH) [22], a somewhat stronger assumption than FPT6= W[1]: it is assumed that there is no 2^o(n) time algorithm forn-variable 3SAT.

Theorem 1.2 ([24]). If there is a functionf and a recursively enumerable class H of hypergraphs with bounded edge size and unbounded treewidth such that CSP(H) can be solved in timef(H)kIk^o(tw(H)/^{log tw(H))}for instancesI with hypergraphH∈ H, then ETH fails.

Unbounded arities. The situation is less understood in the unbounded arity case, i.e., when there is no bound on the maximum edge size inH. First, the complexity in the unbounded-arity case depends on how the constraints are represented. In the bounded-arity case, if each constraint contains at most r variables (r being a fixed constant), then every reasonable representation of a constraint has size|D|^O(r). Therefore, the size of the different representations can differ only by a polynomial factor. On the other hand, if there is no bound on the arity, then there can be exponential difference between the size of succinct representations (e.g., formulas [8]) and verbose representations (e.g., truth tables [26]). The running time of an algorithm is ex- pressed as a function of the input size, hence the complexity of the problem can depend on how the input is represented:

longer representation means that it is potentially easier to obtain a polynomial-time algorithm.

The most well-studied representation of constraints is listing all the tuples that satisfy the constraint. This representation is perfectly compatible with our database-theoretic motivation: the constraints are relations of the database, and a relation is physically stored as a table containing all the tuples in the relation. For this representation, there are classesHwith unbounded treewidth such that CSP restricted to this class is polynomial-time solvable. A trivial example is the classHof all hypergraphs having only a single hyperedge of arbitrary size. There are other classes of hypergraphs with unbounded treewidth such that CSP(H) is solvable in polynomial time: for example, classes with bounded (generalized) hypertree width [16], boundedfractional edge cover number [20], and bounded fractional hypertree width [20, 25]. Thus treewidth is not the right measure for char- acterizing the complexity of the problem.

Our results. We introduce a new hypergraph width measure that we callsubmodular width.Small submodular width means that for every monotone submodular functionbon the vertices of the hypergraphH, there is a tree decomposition where b(B) is small for every bag B of the decomposition.

(This definition makes sense only if we normalize the considered functions: for this reason, we require thatb(e)≤1 for every edgee ofH.) The main result of the paper is showing that bounded submodular width is the property that precisely characterizes the complexity of CSP(H):

Theorem 1.3 (Main). LetHbe a recursively enumerable class of hypergraphs. Assuming ETH, CSP(H) parameterized by the hypergraph H is fixed-parameter tractable if and only ifHhas bounded submodular width.

(3)

Theorem 1.3 has an algorithmic side (algorithm for bounded submodular width) and a complexity side (hardness result for unbounded submodular width). Unlike previous width measures in the literature, where small value of the measure suggests a way of solving CSP(H) it is not at all clear how bounded submodular width is of any help. In particular, it is not obvious what submodular functions have to do with CSP instances. The main idea of our algorithm is that a CSP instance can be “split” into a small number of “uniform” CSP instances; for this purpose, we use a partitioning procedure inspired by a result of Alon et al. [3]. Concep- tually, our algorithm goes beyond previous decomposition techniques in two ways. First, the tree decomposition that we use depends not only on the hypergraph, but on the actual constraint relations in the instance (we remark that this idea first appeared in [26] in a different context that does not directly apply to our problem). Second, we are not only decomposing the set of variables, but we also split the constraint relations. This way, we can apply different decompositions to different parts of the solution space.

The proof of the complexity side of Theorem 1.3 follows the same high-level strategy as the proof of Theorem 1.2 in [24]. In a nutshell, the argument of [24] is the following: if treewidth is large, then there is subset of vertices which is highly connected in the sense that the set does not have a small balanced separator; such a highly connected set implies that there is uniform concurrent flow (i.e., a compatible set of flows connecting every pair of vertices in the set);

the paths in the flows can be used to embed the graph of a 3SAT formula; and finally this embedding can be used to reduce 3SAT to CSP. These arguments build heavily on well-known characterizations of treewidth and results from combinatorial optimization (such as theO(logk) integrality gap of sparsest cut). The proof of Theorem 1.3 follows this outline, but now no such well-known tools are available: we are dealing with hypergraphs and submodular functions in a way that was not explored before in the literature. One of the main difficulties of obtaining Theorem 1.3 is that we have to work in three different domains:

• CSP instances. As our goal is to investigate the existence of algorithms solving CSP, the most obvious domain is CSP instances. In light of previous results, we are especially interested in algorithms based on tree decompositions. For such algorithms, what matters is the existence of subsets of vertices such that restricting the instance to any of these subsets gives an instance with “small” number of solutions. In order to solve the instance, we would like to find a tree decomposition where every bag is such a small set.

• Submodular functions. Submodular width is defined in terms of submodular functions, thus submodular functions defined on hypergraphs is our second natural domain. We need to understand what large submodular width means, that is, what property of the submodular function and the hypergraph makes it impossible to obtain a tree decomposition where every bag has small value.

• Flows and embeddings in hypergraphs. In the hardness proof, our goal is to embed the graph of a 3SAT formula into a hypergraph. Thus we need to define an appropriate notion of embedding and study

what guarantees the existence of embeddings with suit- able properties. As in [24], we use the paths appearing in flows to construct embeddings. For our purposes, the right notion of flow is a collection of weighted paths where the total weight of the paths intersecting each hyperedge is at most 1. This notion of flows has not been studied in the literature before, thus we need to obtain basic tools and results on such flows.

A key question is how to find connections between these domains. As mentioned above and detailed in Section 3, we have a procedure that reduces a CSP instance into a set of uniform CSP instances, and the number of solutions on the different subsets of variables in a uniform CSP instance can be described by a submodular function. This method allows us to move from the domain of CSP instances to the domain of submodular functions. Section 4 is devoted to showing that if submodular width of a hypergraph is large, then there is a certain “highly connected” set in the hypergraph. Highly connected set is defined as a property of the hypergraph and has no longer anything to do with submodular functions. Thus this connection allows us to move from the domain of submodular functions to the study of hypergraphs. In Section 5, we show that a highly connected set in a hypergraph means that graphs can be efficiently embedded into the hypergraph. In particular, the graph of a 3SAT formula can be embedded into the hypergraph, which gives us (as shown in Section 6) a reduction from 3SAT to CSP(H). This connection allows us to move from the domain of embeddings back to the domain of CSP instances.

We can show that bounded submodular width is a strictly more general property than bounded fractional hypertree width, which was previously the most general property known to make CSP(H) FPT (see the full version of the paper).

Thus Theorem 1.3 not only gives a complete characterization of the parameterized complexity of CSP(H), but its algorithmic side proves fixed-parameter tractability in a strictly more general case than what was known before.

2. PRELIMINARIES

Constraint satisfaction problems. We briefly recall the most important notions related to CSP. For more back- ground, see e.g., [18, 12]. An instance of a constraint satisfaction problem is a triple (V, D, C), where: V is a set of variables,Dis a domain of values,Cis a set of constraints, {c1, c2, . . . , cq}. Each constraint ci ∈ C is a pair hsi, Rii, wheresi is a tuple of variables of lengthmi, called thecon- straint scope,andRiis anmi-ary relation overD, called the constraint relation.

For each constrainthsi, Riithe tuples ofRiindicate the allowed combinations of simultaneous values for the variables insi. The lengthmiof the tuplesiis called thearityof the constraint. Asolution to a constraint satisfaction problem instance is a functionffrom the set of variablesV to the domain of valuesDsuch that for each constrainthsi, Riiwith si=hvi1, vi2, . . . , vimi, the tuplehf(vi1), f(vi2), . . . , f(vim)i is a member of Ri. We say that an instance is binary if each constraint relation is binary, i.e., mi= 2 for each constraint. It can be assumed that the instance does not contain two constraints hsi, Rii,hsj, Rji withsi =sj, since in this case the two constraints can be replaced by the constraint hsi, Ri∩Rji. In the input, the relation in a constraint is represented by listing all the tuples of the constraint. We

(4)

denote bykIkthe size of the representation of the instance I= (V, D, C).

LetI= (V, D, C) be a CSP instance and letV⁰⊆V be a nonempty subset of variables. Theprojection pr_V0I ofI to V⁰is a CSPI⁰= (V⁰, D, C⁰), whereC⁰is defined the following way: For each constraintc=h(v1, . . . , vk), Rihaving at least one variable inV⁰, there is a corresponding constraint c⁰ inC⁰. Suppose thatvi1, . . . , vi_` are the variables among v1, . . . , vk that are in V⁰. Then the constraintc⁰ is defined ash(vi1, . . . , vi`), R⁰i, where the relationR⁰is the projection of R to the components i1, . . . , i`, that is, R⁰ contains an

`-tuple (d⁰₁, . . . , d⁰_`) ∈ D^` if and only if there is a k-tuple (d1, . . . , dk)∈Rsuch thatd⁰_j=dij for 1≤j≤`. Clearly, if fis a solution ofI, thenf|V⁰(frestricted toV⁰) is a solution of pr_V0I. For a subsetV⁰ ⊆V, we denote by solI(V⁰) the set of all solutions of pr_V0I. If the instance I is clear from the context, we drop the subscript.

Theprimal graph(orGaifman graph) of a CSP instance I= (V, D, C) is a graph with vertex setV such thatu, v∈V are adjacent if and only if they appear together in the scope of some constraint. Thehypergraphof a CSP instanceI= (V, D, C) is a hypergraphHwith vertex setV, wheree⊆V is an edge if and only if there is a constraint whose scope is e. Let CSP(H) be the problem restricted to instances whose hypergraph is inH.

Paths, separators, and flows in hypergraphs. A pathP in hypergraphHis an ordered sequencev0,v1,. . ., vr of vertices such thatvi and vi−1 are adjacent for every 1≤i < r. We distinguish the endpoints of a path: vertex v0 is thefirst endpointofP andvr is thesecond endpointof P. A path is anX−Y pathif its first endpoint is inX and its second endpoint is inY.

Let H be a hypergraph andX, Y ⊆ V(H) be two (not necessarily disjoint) sets of vertices. An (X, Y)-separator is a set S ⊆ V(H) of vertices such that every X−Y path ofH contains at least one vertex of S. In particular, this means thatX∩Y ⊆S. An assignments:E(H)→R⁺ is a fractional(X, Y)-separatorif everyX−Y pathP iscovered bys, that is, P

e∈E(H),e∩P6=∅s(e) ≥ 1. The weightof the fractional separatorsisP

e∈E(H)s(e).

LetH be a hypergraph and letP be the set of all paths inH. AflowofH is an assignmentf :P →R⁺ such that P

P∈P,P∩e6=∅f(P)≤1 for everye∈E(H). Thevalueof the flowfisP

P∈Pf(P). We say that a pathP appearsin flow f, or simplyP is apath off iff(P)>0. For someX, Y ⊆ V(H), an (X, Y)-flow is a flow f such that only X −Y paths appear inf. A standard LP duality argument shows that the minimum weight of a fractional (X, Y)-separator is equal to the maximum value of an (X, Y)-flow. Iff, f⁰ are flows such that f⁰(P) ≤ f(P) for every path P, then f⁰ is a subflowof f. Thesum of the flowsf1, . . .,fr is a mapping that assigns weightPr

i=1fi(P) to each pathP. If the sum off1,. . .,fr is a flow, then we say thatf1,. . .,fr

arecompatible.

Highly connected sets. An important step in understanding various width measures is showing that if the measure is large, then the (hyper)graph contains a highly connected set (in a certain sense). We define here the notion of highly connected that will be used in the paper. First, recall that a fractional independent setof a hypergraphH is a mappingµ :V(H) →[0,1] such that P

v∈eµ(v) ≤1 for every e ∈ E(H). We extend functions on the vertices

ofHto subsets of vertices of Hthe natural way by setting µ(X) :=P

v∈Xµ(v).

Let µ be a fractional independent set of hypergraphH and letλ >0 be a constant. We say that a setW ⊆V(H) is (µ, λ)-connectedif for any two disjoint setsA, B⊆W, the minimum weight of a fractional (A, B)-separator is at least λ·min{µ(A), µ(B)}. Note that ifWis (µ, λ)-connected, then W is (µ, λ⁰)-connected for everyλ⁰ < λand everyW⁰⊆W is also (µ, λ)-connected. Informally, if W is (µ, λ)-lambda connected for some fractional independent setµ such that µ(W) is “large”, then we callW a highly connected set. For λ >0, we denote by conλ(H) the maximum ofµ(W), taken over all (µ, λ)-connected setWofH. Throughout the paper, λcan be thought of as a sufficiently small universal constant, say, 0.001.

Embeddings. The hardness result presented in the paper and earlier hardness results for CSP(H) [19, 26, 24] are based on embedding some other problem (with a certain graph structure) in a CSP instance whose hypergraph is a member of H. Thus we need appropriate notions of embedding a graph in a (hyper)graph. A crucial difference between the proof of Theorem 1.1 in [19] and the proof of Theorem 1.2 in [24] is that the former result is a based on finding a minor embedding of a grid, while the latter result uses an embedding where the images of distinct vertices are not necessarily disjoint, but can overlap in a controlled way. We define such embeddings the following way. We say that two sets of vertices X, Y ⊆ V(H) touch if either X∩Y 6=∅, or there is an edgee∈E(H) intersecting both X andY. Anembeddingof graphGinto hypergraph His a mapping ψ that maps each vertex of H to a connected subset ofV(G) such that ifuandvare adjacent inG, then ψ(u) andψ(v) touch. Thedepthof a vertexv∈V(H) in em- beddingψisdψ(v) :=|{u∈V(G)|v∈ψ(u)}|, the number of vertices of Gwhose images containv. Thevertex depth of the embedding is maxv∈V(H)dψ(v). Because in our case we want to control the size of the constraint relations, we need a notion of depth that is sensitive to “what the edges see.” We defineedge depthofψto be maxe∈E(H)P

v∈edψ(v).

Equivalently, we can define edge depth as the maximum of P

v∈V(G)|ψ(v)∩e|, taken over all edges ofe ofH.

Trivially, for any graphGand hypergraphH, there is an embedding ofGintoHhaving vertex depth and edge depth at most|V(G)|. IfGhasmedges and no isolated vertices, then|V(G)|is at most 2m. We are interested in how much we can gain compared to this trivial solution of depthO(m).

We define theembedding poweremb(H) to be the maximum (supremum) value ofαfor which there is an integermαsuch that every graphGwithm≥mα edges has an embedding intoHwith edge depthm/α.

Width parametersWe follow the framework of width functions introduced by Adler [1]. Atree decomposition of a hypergraph H is a tuple (T,(Bt)_t∈V_(T₎), where T is a tree and (Bt)_t∈V_(T₎is a family of subsets ofV(H) satisfying the following two conditions: (1) for each e ∈E(H) there is a node t ∈ V(T) such that e ⊆ Bt, and (2) for each v ∈V(H) the set{t ∈V(T) |v∈ Bt} is connected inT. The setsBt are called thebags of the decomposition. Let f: 2^V^(H)→R⁺be a function that assigns a nonnegative real number to each nonempty subset of vertices. The f-width of a tree-decomposition (T,(Bt)t∈V(T)) is max

f(Bt)|t∈ V(T)}. Thef-widthof a hypergraphHis the minimum of thef-widths of all its tree decompositions. In other words,

(5)

f-width(H)≤wif and only if there is a tree decomposition ofH wheref(B)≤wfor every bagB.

The main idea of tree decomposition based algorithms is that if we have a tree decomposition for instance I such that at mostCassignments onBthave to be considered for each bagBt, then the problem can be solved by dynamic programming in time polynomial in C and kIk. The various width notions try to guarantee the existence of such decompositions. For example, the simplest such notion, treewidth, can be defined as tw(H) := s-width(H), where s(B) = |B| −1. Further width notions defined in the literature, such as generalized hypertree width and fractional hypertree width can also be conveniently defined using this setup, see [1].

We generalize the notion off-width from a single function f to a class of functionsF. LetF be an arbitrary (possi- bly infinite) class of functions that assign nonnegative real numbers to nonempty subsets of vertices. TheF-widthof a hypergraphHisF-width(H) := sup

f-width(H)|f∈ F . Thus ifF-width(H)≤k, then for everyf∈ F, hypergraph Hhas a tree decomposition withf-width at mostk. Note that this tree decomposition can be different for the different functionsf. For normalization purposes, we consider only functions f on V(H) that are edge-dominated: f(e) ≤ 1 holds for everye ∈E(H). The main new definition of the paper is submodular width:

Definition 2.1. A function b : 2^V^(H) → R⁺ is submodular ifb(X) +b(Y) ≥b(X∩Y) +b(X ∪Y) holds for every X, Y ⊆ V(H). Given a hypergraph H, let F contain the edge-dominated monotone submodular functions on V(H). Thesubmodular width subw(H)of hypergraphHis F-width(H).

3. FROM CSP INSTANCES TO SUBMODU- LAR FUNCTIONS

We prove the main algorithmic result in this section: CSP(H) is FPT ifHhas bounded submodular width.

Theorem 3.1. LetHbe a class of hypergraphs such that subw(H) ≤ c0 for every H ∈ H. Then CSP(H) can be solved in time2²^O(|V^(H)|)· kIk^O(c⁰⁾.

The proof of Theorem 3.1 is based on two main ideas:

1. A CSP instanceI can be decomposed into a bounded number of “uniform” CSP instancesI1,. . .,It(Lemma 3.9).

Here uniform means that ifB⊆Aare two sets of variables, then every solution of pr_BIj has roughly the same number of extensions to pr_AIj.

2. IfIis a uniform CSP instance, then (the logarithm of) the number of solutions on the different projections of I can be described by an edge-dominated submodular function (Lemma 3.10). Therefore, if the hypergraph HofIhas bounded submodular width, then it follows that there is a tree decomposition where every bag has a small number of solutions.

In the implementation of the first idea (Lemma 3.9), we guarantee uniformity only to subsets of variables that are

“small” in the following hereditary sense:

Definition 3.2. For a CSP instanceI andM ≥1, a set S⊆V isM-smallif|solI(S⁰)| ≤M for every S⁰⊆S.

It is not difficult to find all the M-small sets, and every solution of the instances projected onto these sets:

Lemma 3.3. Let I = (V, D, C) be a CSP instance and M ≥1an integer. There is an algorithm with running time 2^O(|V^|)·poly(kIk, M)that finds the setS of allM-small sets S⊆V and constructssolI(S) for each suchS∈ S.

The following definition gives the precise notion of uniformity that we use:

Definition 3.4. Let I = (V, D, C) be a CSP instance.

For B ⊆ A ⊆ V and an assignment b : B → D, let solI(A|B=b) :={a∈solI(A)|a(x) =b(x)for every x∈B}, the set of all extensions of b to a solution of pr_AI. Let maxI(A|B) = maxb∈sol_I(B)|solI(A|B = b)|. We say that A⊆V isc-uniform(for some integerc) if, for everyB ⊆A, maxI(A|B)≤c|solI(A)|/|solI(B)|.We definemaxI(A|∅) =

|solI(A)|andmaxI(∅|∅) = 1. An instance is(N, c, )-uniform if everyN^c-small set isN-uniform.

Proposition 3.5. For everyB⊆A⊆V andC⊆V, we have

We want to avoid dealing with assignments b ∈ sol(B) that cannot be extended to a member of sol(A) for some A⊇B (that is, sol(A|B=b) =∅). We require that there is no such unextendablebifAandB areM-small:

Definition 3.6. A CSP instance isM-consistentifsol(B) = pr_Asol(A)for all M-small sets B⊆A. A CSP instance is nontrivialifsol({v})6=∅for anyv∈V.

Proposition 3.7. IfIis anM-consistent nontrivial CSP instance, thensol(S)6=∅for every M-small setS.

We can achieveM-consistency by throwing away partial solutions that violate the requirements. InstanceI⁰= (V, D, C⁰) is arefinementofI= (V, D, C) if for everyhs, Ri ∈C, there is ahs, R⁰i ∈C⁰ such thatR⁰⊆R.

Lemma 3.8. Let I = (V, D, C) be a CSP instance and M ≥1an integer. There is an algorithm with running time 2^O(|V^|)·poly(kIk, M) that produces an M-consistent CSP instanceI⁰ that is a refinement ofIwithsol(I) = sol(I⁰).

Our algorithm for decomposing a CSP instance into uniform CSP instances is inspired by a combinatorial result of Alon et al. [3], which shows that, for every fixedn, an n- dimensional point setS can be partitioned into polylog(|S|) O(1)-uniform classes. We follow the same proof idea: the instance is split into two instances if uniformity is violated somewhere, and we analyze the change of a weight function to bound the number of splits performed. However, the parameter setting is different in our proof: we want to partition into f(|V|) classes, but we are satisfied with weaker uniformity. A minor difference is that we require uniformity only onN^c-small sets.

Lemma 3.9. Let I = (V, D, C) be a CSP instance and let N,c be integers and >0. There is an algorithm with running time2²^O(|V^|)^·c/·poly(kIk, N^c)that produces a set of (N, c, )-uniformN^c-consistent nontrivial instancesI1,. . ., It with 0 ≤ t ≤2²^O(|V^|)^·c/, all on the set V of variables, such that

(6)

1. every solution ofI is a solution of exactly oneIi, 2. for every 1≤i≤t, instanceIiis a refinement ofI.

Proof. The main step of the algorithm takes a CSP instance I and either makes it (N, c, )-uniform and N^c- consistent, or splits it into two instancesIsmall, Ilarge. By applying the main step recursively onIsmall and Ilarge, we eventually arrive to a set of (N, c, )-uniform N^c-consistent instances. We will argue that the number of constructed instances is 2²^O(|V^|)^·c/.

In the main step, we first check if the instance is trivial;

in this case we can stop witht= 0. Otherwise, we invoke the algorithm of Lemma 3.8 to obtain anN^c-consistent refinement of the instance, without losing any solution. Next we check if thisN^c-consistent instanceIis (N, c, )-uniform.

This can be tested in time 2^O(|V^|)·poly(kIk, N^c) if we use Lemma 3.3 to find all theN^c-small sets and the corresponding sets of solutions. Suppose thatN^c-small setsB⊆Avi- olate uniformity, that is, max(A|B)> M|sol(A)|/|sol(B)|. Let solsmall(B) contain those tuplesbfor which|sol(A|B= b)| ≤ √

M|sol(A)|/|sol(B)| and let sollarge(B) = sol(B)\ solsmall(B). Note that

|sol(A)| ≥ |sollarge(B)| ·(√

M|sol(A)|/|sol(B)|) (as every tupleb∈sollarge(B) has at least√

M|sol(A)|/|sol(B)| extensions toA), hence |sollarge(B)| ≤ |sol(B)|/√

M. Let instanceIsmall (resp.,Ilarge) be obtained fromI by adding the constrainthB,solsmall(B)i(resp.,hB,sollarge(B)i). Clearly, the set of solutions ofI is the disjoint union of the sets of solutions ofIsmallandIlarge. This completes the description of the main step.

It is clear that if the recursive procedure stops, then the instances at the leaves of the recursion satisfy the two requirements. We show that the height of the recursion tree can be bounded from above by a functionh(|V|, c, ) depend- ing only on|V|,c, and; in particular, this shows that the recursive algorithm eventually stops and produces at most 2^h(|V^|,c,)instances.

Let us consider a path in the recursion tree starting at the root, and letI¹, I², . . ., I^p be the corresponding N^c consistent instances. If a set S is N^c-small inI^j, then it is N^c-small in I^j⁰ for every j⁰ > j: the main step cannot increase|sol(S)|for anyS. Thus, with the exception of at most 2^|V^| values ofj, instancesI^j andI^j+1 have the same N^c-small sets. Let us consider a subpathI^x,. . .,I^y such that all these instances have the sameN^c-small sets. We show that the length of this subpath is at mostO(3^|V^|·c/), hencep=O(2^|V^|·3^|V^|·c/).

For the instanceI^j, let us define the following weight:

W^j = X

∅⊆B⊆A⊆V A, BareN^c-small

log max_Ij(A|B).

We bound the length of the subpathI^x,. . .,I^yby analyzing how this weight changes inIlarge andIsmall compared toI. Note that 0≤W^j ≤3^|V^|logN^c = 3^|V^|·clogN: the sum consists of at most 3^|V^|terms and (asAisN^c-small and the instanceI^j isN^c consistent and nontrivial) max_Ij(A|B) is between 1 andN^c. We show thatW^j+1≤W^j−(/2) logN, which immediately implies that the length of the subpath is O(3^|V^|·c/). Let us inspect howW^j+1changes compared to W^j. SinceI^jandI^j+1have the sameN^c-small sets, no new

term can appear in W^j+1. It is clear that max_Ii+1(A|B) cannot be greater than max_Ii(A|B) for anyA, B. However, there is at least one term that strictly decreases. Suppose first thatI^j+1was obtained fromI^jby adding the constraint hB,solsmall(B)i. Then

log max_Ij+1(A|B)≤log√

N|sol_Ij(A)|

|sol_Ij(B)|

≤log(max_Ij(A|B)/√

N) = log max_Ij(A|B)−(/2) logN.

On the other hand, ifI^j+1was obtained by adding the con- strainthB,sollarge(B)i, then

log max_Ij+1(B|∅) = log|sol_Ij+1(B)|

≤log(|sol_Ij(B)|/√

N) = log max_Ij(B|∅)−(/2) logN.

In both cases, we get that at least one term decreases by at least (/2) logN.

Assume for a moment that we have a 1-uniform instance Iwith hypergraphH. Note that by Prop 3.5(1), this means that max(A|B) = |sol(A)|/|sol(B)|. Suppose that every constraint contains at most N tuples and let us define the function b(S) = log_N|sol(S)|. For every edge e ∈ E(H), there is a corresponding constraint, which has at most N tuples by the definition ofN. Thus|sol(e)| ≤N and hence b(e)≤1 for everye∈ E(H), that is,bis edge dominated.

The crucial observation of this section is that this function bis submodular:

b(X) +b(Y)

= log_N|sol(X)|+ log_N

|sol(X∩Y)| |sol(Y)|

|sol(X∩Y)|

= log_N

|sol(X)||sol(X∪Y)|

|sol(X)|

+ log_N|sol(X∩Y)|

=b(X∩Y) +b(X∪Y) (the equalities follow from 1-uniformity; the inequality uses Prop. 3.5(2) withA=Y,B=X∩Y,C=X). Therefore, if subw(H) ≤ c, then H has a tree decomposition where b(B)≤c and hence|sol(B)| ≤ N^c for every bagB. This allows us to solve the problem by dynamic programming in time polynomial inN^c.

Lemma 3.9 guarantees some uniformity for the created instances, but not perfect 1-uniformity and only for theN^c- small sets. In Lemma 3.10, we definebin a slightly different way: we add small terms to correct errors arising from weaker uniformity and we truncate the function at large values (i.e., for sets that are notN^c-small).

Lemma 3.10. LetI = (V, D, C) be a CSP instance with hypergraph H such that |sol(e)| ≤ N for every e∈ E(H).

IfI isN^c-consistent and(N, c, ³)-uniform for some c≥1 and := 1/|V|, then the following function b is an edge- dominated, monotone, submodular function on V(H):

b(S) :=

((1−)log_N|sol(S)|+ 2²|S| −³|S|² ifS isN^c-small, (1−)c+ 2²|S| −³|S|² otherwise.

(7)

If subw(H) ≤ (1−)c, then there is a tree decomposition where every bag isN^c-small, and we can use this tree decomposition to find a solution. In fact, in this caseN^c- consistency implies that every nontrivial instance has a solution. Putting together this observation with Lemmas 3.9 and 3.10, the proof of Theorem 3.1 follows.

4. FROM SUBMODULAR FUNCTIONS TO HIGHLY CONNECTED SETS

The aim of this section is to show that if a hypergraph Hhas large submodular width, then there is a large highly connected set inH. The main result is the following:

Theorem 4.1. For every sufficiently small constantλ >

0, the following holds. Letbbe an edge-dominated monotone submodular function ofH. If theb-width ofHis greater than

3

2(w+ 1), thenconλ(W)≥w.

For the proof of Theorem 4.1, we need to show that if there is no tree decomposition whereb(B) is small for every bag B, then a highly connected set exists. There is a standard recursive procedure that either builds a tree decomposition or finds a highly connected set (see e.g., [13, Section 11.2]).

Simplifying somewhat, the main idea is that if a the graph can be decomposed into smaller graphs by splitting a certain set of vertices into two parts, then a tree decomposition for each part is constructed using the algorithm recursively, and the tree decompositions for the parts are joined in an appropriate way to obtain a tree decomposition for the original problem. On the other hand, if the set of vertices cannot be split, then we can conclude that it is highly connected.

This high-level idea has been applied for various notions of tree decompositions [30, 28, 2, 29], and it turns out to be useful in our context as well. However, we need to overcome two major difficulties:

1. Highly connected set in our context is defined as not having certainfractional separators(i.e., weight assignments). However, if we want to build a tree decomposition in a recursive manner, we needinteger separators (i.e., subsets of vertices) that decompose the hypergraph into smaller parts.

2. Measuring the sizes of sets with a submodular function bcan lead to problems, since the size of the union of two sets can be much smaller the sum of the sizes of the two sets. We need the property that, roughly speaking, removing a “large” part from a set makes it “much smaller.” For example, ifAandB are components of H\S, and bothb(A) andb(B) are large, then we need the property that both of them are much smaller than b(A∪B). Adler [1, Section 4.2] investigates the relation between some notion of highly connected set and f-width, but assumes that f is additive: ifA and B do not touch, thenf(A∪B) =f(A) +f(B). However, for a submodular functionb, it very well may be that b(A) =b(B) =b(A∪B).

To overcome the first difficulty, we have to understand what fractional separation really means. We show that if there is a fractional (A, B)-separator of weightw, then forevery edge-dominated monotone submodular functionb, there is an (A, B)-separatorSwithb(S) =O(w). Note that this sep- aratorS can be different for different functionsb, so we are

not claiming that there is a single (A, B)-separator S that is small in everyb. The converse is also true (we omit the proof), thus this gives a novel characterization of fractional separation, tight up to a constant factor. This result is the key idea that allows us to move from the domain of submodular functions to the domain of pure hypergraph properties:

if there is no (A, B)-separator such thatb(S) is small, then we know that there is no small fractional (A, B)-separator, which is a property of the hypergraphH only and has no longer anything to do with the submodular functionb.

To overcome the second difficulty, we introduce a transformation that turns a monotone submodular functionbon V(H) into a function b^∗ that encodes somehow the neigh- borhood structure ofH as well. The new functionb^∗is no longer monotone and submodular, but is has a number of remarkable properties, for example, b^∗ remains edge dominated and b^∗(S)≥b(S) for every setS ⊆V(H), implying thatb^∗-width is not smaller thanb-width. The main idea is to prove Theorem 4.1 for b^∗-width instead ofb-width. Be- cause of the wayb^∗encodes the neighborhoods, the second difficulty will disappear: for example, it will be true that b^∗(A∪B) =b^∗(A) +b^∗(B) if there are no edges between A and B, that is, b^∗ is additive on disjoint components.

Lemma 4.3 formulates (in a somewhat technical way) the exact property of b^∗ that we will need. It turns out that the result mentioned in the previous paragraph remains true withbreplaced byb^∗: if there is a fractional (A, B)-separator of weight w, then there is an (A, B)-separatorS such that not onlyb(S), but evenb^∗(S) isO(w).

The functionb^∗. We define the functionb^∗ the following way. LetH be a hypergraph and letb be a monotone submodular function defined onV(H). LetS_V_(H)be the set of all permutations ofV(H). For a permutationπ∈SV(H), letN_π⁻(v) be the neighbors ofvprecedingvin the ordering π. Forπ∈SV(H) andZ ⊆V(H), we define

∂bπ,Z(v) :=b(v∪(Nπ⁻(v)∩Z))−b(Nπ⁻(v)∩Z).

In other words, ∂bπ,Z(v) is the marginal value of v with respect to the set of its neighbors in Z preceding it. We abbreviate ∂bπ,V(H) by ∂bπ. We extend the definition to subsets by∂bπ,Z(S) :=P

v∈S∂bπ,Z(v). Next, we define bπ(Z) :=∂bπ,Z(Z) =X

v∈Z

∂bπ,Z(v) b^∗(Z) := min

π∈S_V_(H)bπ(Z).

Thus bπ(Z) is the sum of the marginal values with respect to a given ordering, whileb^∗(Z) is the smallest possible sum taken over all possible orderings.

Proposition 4.2. Let H be a hypergraph and letb be a monotone submodular function defined onV(H). For every π∈S_V_(H) andZ⊆V(H)we have

1. bπ(Z)≥b(Z), 2. b^∗(Z)≥b(Z), 3. bπ(Z) =b(Z)

ifZ is a clique,

4. ∂bπ,Z1(v)≤∂bπ,Z2(v) if Z2⊆Z1,

5. ∂bπ(v)≤∂bπ,Z(v), 6. b^∗(X ∪Y) ≤ b^∗(X) +

b^∗(Y).

The following property ofb^∗allows us to avoid the second difficulty described at the beginning of Section 4.

Lemma 4.3. LetHbe a hypergraph, let bbe a monotone submodular function defined onV(H)and letW be a set of

(8)

vertices. LetπW be an ordering of V(H), and let µ(v) :=

∂bπ_W,W(v) forv∈W andµ(v) = 0otherwise. LetA, B⊆ W be two disjoint sets, and letS be an(A, B)-separator. If b^∗(S)< µ(A), µ(B), thenb^∗((C∩W)∪S)< µ(W)for every componentC ofH\S.

Next we show that having a small fractional (A, B)-separator means that for every monotone edge-dominated submodular functionb, there is an (A, B)-separatorSsuch thatb^∗(S) (and henceb(S)) is small. The proof is based on a standard trick that is often used for rounding fractional solutions for separation problems: we define a distance function and show by an averaging argument that cutting at some distancet gives a small separator. However, in our setting, we need significant new ideas to make this trick work: the main difficulty is that the cost function b is defined on subsets of vertices and is not a function defined by the cost of vertices.

To overcome this problem, we use the function ∂bπ(v) to assign a cost to every single vertex.

Theorem 4.4. LetHbe a hypergraph,X, Y ⊆V(H)two sets of vertices, and b : V(H) → R⁺ an edge-dominated monotone submodular function. Suppose that s is a fractional(X, Y)-separator of weight at most w. Then there is an(X, Y)-separator S⊆V(H)withb^∗(S) =O(w).

Proof. Let us definex(v) := max{1,P

e∈E(H),v∈es(e)}. It is clear that ifPis a path fromXtoY, thenP

v∈Px(v)≥ 1. We define the distanced(v) to be the minimum ofP

v⁰∈Px(v⁰), taken over all paths fromX tov(this means thatd(v)>0 is possible for some v ∈ X). It is clear that d(v) ≥ 1 for every v ∈ Y. Let us associate the closed interval ι(v) = [d(v)−x(v), d(v)] to each vertex v. Ifv is in X, then the left endpoint ofι(v) is 0, while ifv is inY, then the right endpoint ofι(v) is at least 1. The following claim is easy to see:

Claim 1: Ifuandvare adjacent, thenι(u)∩ι(v)6=∅. Theclassof a vertexv∈V(H) is the largest integerκ(v) such thatx(v)≤2^−κ(v), and we defineκ(v) :=∞ifx(v) = 0. Recall that x(v) ≤ 1, thus κ(v) is nonnegative. The offsetof a vertexv is the unique value 0≤α < 2·2^−κ(v) such thatd(v) =i(2·2^−κ(v)) +αfor some integeri. Let us define an orderingπ= (v1, . . . , vn) ofV(H) such thatκ(v) is nondecreasing, and among vertices having the same class, the offset is nondecreasing.

Let directed graph D be the orientation of the primal graph ofH such that if vi and vj are adjacent and i < j, then there is a directed edge−−→vivj inD. If P is a directed path inD, then thewidthofP is the length of the interval S

v∈Pι(v) (note that by Claim 1, this union is indeed an interval). The following claim bounds the maximum possible width of a directed path:

Claim 2: IfP is a directed pathD starting atv, then the width ofP is at most16x(v).

To prove Claim 2, we first show that if every vertex of P has the same classκ(v), then the width ofP is at most 4·2^−κ(v). Since the class is nondecreasing along the path, we can partition the path into subpaths such that every vertex in a subpath has the same class and the classes are distinct on the different subpaths. The width of P is at most the sum of the widths of the subpaths, which is at mostP

i≥κ(v)4·2⁻ⁱ= 8·2^−κ(v)≤16x(v).

Suppose now that every vertex of P has the same class κ(v) as the first vertexvand leth:= 2^−κ(v). As the offset

is nondecreasing, pathPcan be partitioned into two parts: a subpathP1containing vertices with offset at leasth, followed by a subpathP2 containing vertices with offset less than h (one ofP1 andP2 can be empty). We show that each ofP1

andP2 has width at most 2h, which implies that the width ofP is at most 4h. Observe that ifv∈P1andι(v) contains a pointi·2hfor some integeri, then, consideringx(v)≤h and the bounds on the offset of v, this is only possible if ι(v) = [i·2h, i·2h+h], i.e.,i·2his the left endpoint ofι(v).

Thus if I1 =S

v∈P1ι(v) contains i·2h, then it is the left endpoint ofI1. Therefore, I1 can containi·2hfor at most one value ofi, which immediately implies that the length of I1 is at most 2h. We can argue similarly for P2.

Claim 3: P

v∈V(H)x(v)c(v)≤wforc(v) :=∂bπ(v).

Let us examine the contribution of an edgee∈E(H) with values(e) to the sum. For everyv∈e, edgeeincreases the valuex(v) by at mosts(e). Thus the total contribution ofe is at most

s(e)·X

v∈e

c(v) =s(e)·X

v∈e

∂bπ(v)≤s(e)·X

v∈e

∂bπ,e(v)

=s(e)bπ(e)≤s(e)b(e)≤s(e), where the first inequality follows Prop. 4.2(5); the second inequality follows form Prop. 4.2(3); the last inequality follows from the fact that b is edge dominated. Therefore, P

v∈V(H)x(v)c(v)≤P

e∈E(H)s(e)≤w, proving Claim 3.

We defineC(S) to be the set of all vertices from which a vertex ofS is reachable on a directed path inD.

Claim 4: For everyS⊆V(H),P

v∈C(S)c(v) =bπ(C(S)).

Observe that for anyv ∈ C(S), every inneighbor of vis also inC(S), henceNπ⁻(v)⊆ C(S). Therefore,∂bπ,C(S)(v) =

∂bπ(v) =c(v) and Claim 4 follows.

Now we are ready to prove the lemma. LetS(t) be the set of all vertices v ∈ V(H) for which t ∈ ι(v) and let SC(t) = C(S(t)). Observe that for every 0 ≤ t ≤ 1, the setS(t) (and henceSC(t)) separatesX fromY. We use an averaging argument to show that there is a 0 ≤ t≤ 1 for whichbπ(SC(t)) isO(w). Asb^∗(SC(t))≤bπ(SC(t)), the set SC(t) satisfies the requirement of the lemma.

If we can show thatR1

0 bπ(SC(t))dt=O(w), then the existence of the required tfollows. LetIv(t) = 1 ifv∈SC(t) and letIv(t) = 0 otherwise. IfIv(t) = 1, then there is a path P inD fromvtoS(t). By Claim 2, the width of this path is at most 16x(v), thus t ∈ [d(v)−16x(v), d(v) + 15x(v)].

Therefore,R1

0 Iv(t)dt≤31x(v). Now we have Z 1

0

bπ(SC(t))dt= Z 1

0

X

v∈SC(t)

c(v)dt

= Z1

0

X

v∈V(H)

c(v)Iv(t)dt= X

v∈V(H)

c(v) Z 1

0

Iv(t)dt

≤31 X

v∈V(H)

x(v)c(v)≤31w (we used Claim 4 in the first equality and Claim 3 in the last inequality).

By Prop 4.2(2), the following lemma implies Theorem 4.1.

The proof is a recursive procedure finding a tree decomposition, using Lemmas 4.3 and 4.4 to overcome the difficulties described at the beginning of the section.

(9)

Lemma 4.5. Let b be an edge-dominated monotone submodular function of H. Ifb^∗-width(H) > ³₂(w+ 1), then conλ(W)≥w(for some universal constantλ).

5. FROM HIGHLY CONNECTED SETS TO EMBEDDINGS

We show that the existence of highly connected sets implies that the hypergraph has large embedding power:

Theorem 5.1. For every sufficiently smallλ >0and hy- pergraphH, there is a constantmH,λ such that every graph Gwithm≥mH,λedges has an embedding intoHwith edge depthO(m/(λ³²conλ(H)¹⁴)).

Our strategy is similar to [24]: we show that a highly connected set implies that a uniform concurrent flow exists; the paths appearing in the uniform concurrent flow can be used to embed the line graph of a complete graph; and every graph has an appropriate embedding in the line graph of a complete graph. To make this strategy work, we need gener- alizations of concurrent flows, multicuts, and multicommodity flows in our hypergraph setting and we need to obtain results that connect these concepts to highly connected sets.

LetWbe a set of vertices and let (X1, . . . , Xk) be a partition ofW. Auniform concurrent flow of valueon(X1, . . . , Xk) is a compatible set of ^k₂

flows Fi,j (1≤i < j≤k) where Fi,j is an (Xi, Xj)-flow of value. We will need a uniform concurrent flow connecting a set of cliques, thus our first goal is to find a highly connected set that can be partitioned intokcliques in an appropriate way. (Proofs of this section appear in the full version of the paper.)

Lemma 5.2. LetHbe a hypergraph and let0< λ <1/16 be a constant. Then there is fractional independent setµ, a(µ, λ/6)-connected setW, and a partition(K1, . . . , Kk)of W such thatk= Ω(λp

conλ(H)), and for every1≤i≤k, Ki is a clique withµ(Ki)≥1/2.

IfHis connected, then the maximum value of a uniform concurrent flow on (X1, . . . , Xk) is at least 1/ ^k₂

= Ω(k⁻²):

if each of the ^k₂

flows has value 1/ ^k₂

, then they are clearly compatible. The following lemma shows that ifX1,. . .,Xr

are appropriate cliques of a (µ, λ)-connected set, then we can guarantee a better bound Ω(k⁻³²).

Lemma 5.3. LetH be a hypergraph, µa fractional independent set ofH, andW ⊆V(H)a(µ, λ)-connected set of W for some 0 < λ < 1. Let (K1, . . . , Kk) be a partition of W such that Ki is a clique and µ(Ki) ≥ 1/2 for every 1≤i≤k. Then there is a uniform concurrent flow of value Ω(λ/k³²)on(K1, . . . , Kk).

Intuitively, the intersection structure of the paths appearing in a uniform concurrent flow on cliquesK1, . . ., Kr is reminiscent of the edges of the complete graph onrvertices:

if{i1, j1} ∩ {i2, j2} 6= ∅, then every path ofFi1,j1 touches every path ofFi2,j2. We use the following result from [24]

(restated using the terminology of this paper), which shows that line graphs of cliques have good embedding properties.

LetLkbe the line graph of thek-clique.

Lemma 5.4. For everyk >1there is a constant nk>0 such that for every G(V, E) with |E|> nk and no isolated vertices, the graphGhas an embedding intoLk with vertex depthO(|E|/k²).

The proof of Theorem 5.1 uses Lemmas 5.2 and 5.3 to find highly connected set and a uniform concurrent flow on it, and then uses Lemma 5.4 to embed the vertices ofGinto the paths appearing in the flows.

6. FROM EMBEDDINGS TO HARDNESS OF CSP

We prove the main hardness result in this section:

Theorem 6.1. Let H be a recursively enumerable class of hypergraphs with unbounded submodular width. If there is an algorithm Aand a function f such that Asolves every instanceIof CSP(H) with hypergraphH∈ Hin timef(H)· kIk^o(subw(H)

1

4), then the Exponential Time Hypothesis fails.

In particular, Theorem 6.1 implies that CSP(H) for such a His not fixed-parameter tractable:

Corollary 6.2. IfHis a recursively enumerable class of hypergraphs with unbounded submodular width, then CSP(H) is not fixed-parameter tractable, unless ETH fails.

The Exponential Time Hypothesis (ETH) states that there is no 2^o(n) time algorithm forn-variable 3SAT. The Sparsi- fication Lemma of Impagliazzo, Paturi, and Zane [22] shows that ETH is equivalent to the assumption that there is no algorithm for 3SAT whose running time is subexponential in the number of clauses. This result will be crucial for our hardness proof, as our reduction from 3SAT is sensitive to the number of clauses.

To prove Theorem 6.1, we show that a “too fast” algorithm A implies that a subexponential-time algorithm for 3SAT exists. We use the characterization of submodular width from Section 4 and the embedding results of Section 5 to reduce 3SAT to CSP(H) by embedding the incidence graph of a 3SAT formula into a hypergraphH∈ H. The basic idea of the proof is that if the 3SAT formula hasmclauses and the edge depth of the embedding ism/r, then we can gain a factorrin the exponent of the running time. If submodular width is unbounded in H, then we can make this gap r between the number of clauses and the edge depth arbitrary large, and hence the exponent can be arbitrarily smaller than the number of clauses, i.e., the algorithm is subexponential in the number of clauses.

The following simple lemma gives a transformation that turns a 3SAT instance into a binary CSP instance.

Lemma 6.3 ([24]). Given an instance of 3SAT withn variables andmclauses, it is possible to construct in polynomial time an equivalent CSP instance withn+mvariables, 3mbinary constraints, and domain size3.

Next we show that an embedding from graphGto hyper- graphHcan be used to simulate a binary CSP instanceI1

having primal graphGby a CSP instance I2 whose hypergraph is H. The size of the constraint relations of I2 can grow very large: the edge depth of the embedding deter- mines how large is this increase.

Lemma 6.4. Let I1 = (V1, D1, C1) be a binary CSP instance with primal graphGand letφ be an embedding ofG into a hypergraph H with edge depth q. Given I1, H, and the embeddingφ, it is possible to construct (in time polynomial in the size of theoutput) an equivalent CSP instance I2= (V2, D2, C2)with hypergraphH where the size of every constraint relation is at most|D1|^q.

(10)

With these tools at hand, we can prove Theorem 6.1 the following way. We show that if there is a classHof hypergraphs with unbounded submodular width and an algorithm A as in Theorem 6.1, then this algorithm can be used to solve 3SAT in subexponential time. The main ingredients are the embedding result of Theorem 5.1, and Lemmas 6.3 and 6.4 above on reduction to CSP. Furthermore, we need a way of choosing an appropriate hypergraph from the setH. As discussed earlier, the larger the submodular width of the hypergraph is, the more we gain in the running time. How- ever, we should not spend too much time on constructing the hypergraph and on finding an embedding. Therefore, we use the same technique as in [24]: we enumerate a certain number of hypergraphs and we try all of them simultane- ously. The number of hypergraphs enumerated depends on the size of the 3SAT instance. This will be done in such a way that guarantees that we do not spend too much time on the enumeration, but eventually every hypergraph inH is considered for sufficiently large input sizes.

7. REFERENCES

[1] I. Adler.Width functions for hypertree decompositions.

PhD thesis, Albert-Ludwigs-Universit¨at Freiburg, 2006.

[2] I. Adler, G. Gottlob, and M. Grohe. Hypertree width and related hypergraph invariants.European J.

Combin., 28(8):2167–2181, 2007.

[3] N. Alon, I. Newman, A. Shen, G. Tardos, and N. Vereshchagin. Partitioning multi-dimensional sets in a small number of “uniform” parts.European J.

Combin., 28(1):134–144, 2007.

[4] C. Beeri, R. Fagin, D. Maier, A. O. Mendelzon, J. D.

Ullman, and M. Yannakakis. Properties of acyclic database schemes. InSTOC, pages 355–362, 1981.

[5] C. Beeri, R. Fagin, D. Maier, and M. Yannakakis. On the desirability of acyclic database schemes.J. Assoc.

Comput. Mach., 30(3):479–513, 1983.

[6] A. K. Chandra and P. M. Merlin. Optimal implementation of conjunctive queries in relational data bases. InSTOC, pages 77–90, 1977.

[7] C. Chekuri and A. Rajaraman. Conjunctive query containment revisited.Theoret. Comput. Sci., 239(2):211–229, 2000.

[8] H. Chen and M. Grohe. Constraint satisfaction problems with succinctly specified relations, 2006.

Manuscript. Preliminary version inDagstuhl Seminar Proceedings 06401: Complexity of Constraints.

[9] F. R. K. Chung, R. L. Graham, P. Frankl, and J. B.

Shearer. Some intersection theorems for ordered sets and graphs.J. Combin. Theory Ser. A, 43(1):23–37, 1986.

[10] R. G. Downey and M. R. Fellows.Parameterized Complexity. Monographs in Computer Science.

Springer, New York, 1999.

[11] R. Fagin. Degrees of acyclicity for hypergraphs and relational database schemes.J. Assoc. Comput.

Mach., 30(3):514–550, 1983.

[12] T. Feder and M. Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory.SIAM J. Comput., 28(1):57–104, 1999.

[13] J. Flum and M. Grohe.Parameterized Complexity Theory. Springer, Berlin, 2006.

[14] E. C. Freuder. Complexity of k-tree structured constraint satisfaction problems. InProc. of AAAI-90, pages 4–9, Boston, MA, 1990.

[15] G. Gottlob, N. Leone, and F. Scarcello. Hypertree decompositions and tractable queries.Journal of Computer and System Sciences, 64:579–627, 2002.

[16] G. Gottlob, F. Scarcello, and M. Sideri.

Fixed-parameter complexity in AI and nonmonotonic reasoning. Artificial Intelligence, 138(1-2):55–86, 2002.

[17] G. Gottlob and S. Szeider. Fixed-parameter algorithms for artificial intelligence, constraint satisfaction and database problems.The Computer Journal, 51(3):303–325, 2008.

[18] M. Grohe. The structure of tractable constraint satisfaction problems. InMFCS’06, pages 58–72, 2006.

[19] M. Grohe. The complexity of homomorphism and constraint satisfaction problems seen from the other side.J. ACM, 54(1):1, 2007.

[20] M. Grohe and D. Marx. Constraint solving via fractional edge covers. InSODA’06, pages 289–298, 2006.

[21] M. Grohe and D. Marx. On tree width, bramble size, and expansion.J. Combin. Theory Ser. B,

99(1):218–228, 2009.

[22] R. Impagliazzo, R. Paturi, and F. Zane. Which problems have strongly exponential complexity? J.

Comput. System Sci., 63(4):512–530, 2001.

[23] P. G. Kolaitis and M. Y. Vardi. Conjunctive-query containment and constraint satisfaction.J. Comput.

Syst. Sci., 61(2):302–332, 2000.

[24] D. Marx. Can you beat treewidth? To appear in Theory of Computing.

[25] D. Marx. Approximating fractional hypetree width. In SODA’09, pages 902–911, 2009.

[26] D. Marx. Tractable structures for constraint satisfaction with truth tables. InSTACS’09, pages 649–660, 2009.

[27] R. Niedermeier.Invitation to fixed-parameter algorithms, volume 31 ofOxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2006.

[28] S. Oum. Approximating rank-width and clique-width quickly. InWG’05, pages 49–58, 2005.

[29] S. Oum and P. Seymour. Testing branch-width.J.

Combin. Theory Ser. B, 97(3):385–393, 2007.

[30] S.-i. Oum and P. Seymour. Approximating clique-width and branch-width.J. Combin. Theory Ser. B, 96(4):514–528, 2006.

[31] F. Scarcello, G. Gottlob, and G. Greco. Uniform constraint satisfaction problems and database theory.

InComplexity of Constraints, pages 156–195, 2008.

[32] M. Yannakakis. Algorithms for acyclic database schemes. InVLDB, pages 82–94, 1981.